Hazard models are powerful distress models.
Popular since Shumway (2001).
\(y_{it}\)s are binary distress indicator, \(\vec x_{it}\)s are firm covariates, and \(\vec{z}_t\)s are macro variables.
E.g., see Chava and Jarrow (2004), Beaver, McNichols, and Rhie (2005), and Campbell and Szilagyi (2008).
Cannot be captured by firm specific covariates, \(\vec x_{it}\), and macro covariates, \(\vec z_t\). Major issue for portfolio risk.
Extend to include latent effects
\[ \begin{align*} g(E(y_{it}|\vec x_{it}, \vec z_t, \vec u_t, \vec o_{it})) &= \vec\beta^\top\vec x_{it}+ \vec \gamma^\top \vec z_t + \vec u_t^\top \vec o_{it} \\ \vec u_t &\sim p_{\vec \theta}(\vec u_{t-1}) \end{align*} \]
Pioneered by Duffie et al. (2009).
E.g., see Koopman, Lucas, and Schwaab (2011), Hwang (2012), Chen and Wu (2014), Qi, Zhang, and Zhao (2014), Nickerson and Griffin (2017), Azizpour, Giesecke, and Schwenkler (2018), and Kwon and Lee (2018) for other papers on time-varying effects.
Only a frailty model as in Vaupel, Manton, and Stallard (1979) for certain link functions where \(\vec u_t^\top \vec o_{it}\) is a multiplicative factor on the instantaneous hazard. This is not the case for e.g., the logit link function where \(\vec u_t^\top \vec o_{it}\) is a multiplicative factor on the odds.
Although an OU process is a reasonable starting model for the frailty process, one could allow much richer frailty models. …, however, we have found that even our relatively large data set is too limited to identify much of the time-series properties of frailty. … For the same reason, we have not attempted to identify sector-specific frailty effects.
Duffie et al. (2009)
Data set.
Models without frailty.
Models with frailty.
Merge CRSP and Compustat.
Merge with Moody’s Analytics Default & Recovery Database.
At risk: has not had a distress, is after resolution date, or 12 months after distress if resolution date is missing.
Events are alternative dispute resolution, conservatorship, cross default, loan forgiven, chapter 7, placed under administration, seized by regulators, deposit freeze, suspension of payments, grace period default, payment moratorium, prepackaged chapter 11, indenture modified, bankruptcy, receivership, missed principal payment, missed principal and interest payments, distressed exchange, chapter 11, and missed interest payment.
Missed interest payment is by far the most common.
Matching to Compustat is done with CUSIPs, ticker symbols, and company names.
Use quatlery data and impute yearly data when missing.
Data is lagged with 3 months and carried forward for up to 1 year.
Used to compute past 1 year excess log return.
Used to compute log relative market size.
Data is lagged with 1 month and carried forward for up to 3 months.
Very good predictor (Bharath and Shumway 2008).
Require 1 year initial data and at least 3 months of data in each window.
siccd if available. Otherwise use CRSP sich.
Final samples has 624,692 firm-months, 4,433 firms and 721 events.
Included variables are
All winsorized at 1% and 99%.
| Df | AIC | |
|---|---|---|
| First model | 12 | 7612.93 |
| New denominator | 12 | 7559.93 |
| Add distance-to-default | 13 | 7469.90 |
| Add macro variables | 15 | 7447.24 |
| Simplify model | 11 | 7441.04 |
| Df | AIC | |
|---|---|---|
| First model | 12 | 7612.93 |
| New denominator | 12 | 7559.93 |
| Add distance-to-default | 13 | 7469.90 |
| Add macro variables | 15 | 7447.24 |
| Simplify model | 11 | 7441.04 |
Issue with small denominators as in Campbell and Szilagyi (2008). Change denominator to 50% total assets and 50% market value.
| Df | AIC | |
|---|---|---|
| First model | 12 | 7612.93 |
| New denominator | 12 | 7559.93 |
| Add distance-to-default | 13 | 7469.90 |
| Add macro variables | 15 | 7447.24 |
| Simplify model | 11 | 7441.04 |
| Df | AIC | |
|---|---|---|
| First model | 12 | 7612.93 |
| New denominator | 12 | 7559.93 |
| Add distance-to-default | 13 | 7469.90 |
| Add macro variables | 15 | 7447.24 |
| Simplify model | 11 | 7441.04 |
Past 1 year market log return and 1 year T-bill rate as in Duffie et al. (2009) and other papers .
| Df | AIC | |
|---|---|---|
| First model | 12 | 7612.93 |
| New denominator | 12 | 7559.93 |
| Add distance-to-default | 13 | 7469.90 |
| Add macro variables | 15 | 7447.24 |
| Simplify model | 11 | 7441.04 |
| Estimate | Z-stat | |
|---|---|---|
| Intercept | -8.448 | -17.985 |
| Distance-to-default | -1.929 | -9.034 |
| Log excess return | -0.713 | -18.711 |
| Total liabilities / size | 0.437 | 8.365 |
| T-bill rate | -0.281 | -5.113 |
| Net income / size | -0.110 | -5.650 |
| Relative log market size | -0.107 | -1.702 |
| Current ratio | -0.101 | -1.999 |
| Idiosyncratic volatility | 0.092 | 2.449 |
| Log market return | 0.078 | 2.287 |
| Retained Earnings / size | 0.042 | 1.543 |
Terms that are included in Duffie et al. (2009 table III) have similar sign.
| AIC | Df | LR-stat | P-value | |
|---|---|---|---|---|
| Full model | 7377.98 | |||
| % Relative log market size | 7378.14 | 1 | 2.16 | 0.141196 |
| % Retained Earnings / size | 7378.98 | 1 | 3.00 | 0.083270 |
| % Log market return | 7380.39 | 1 | 4.41 | 0.035750 |
| % Current ratio | 7382.43 | 1 | 6.45 | 0.011094 |
| % Net income / size (spline term) | 7393.20 | 3 | 21.23 | 0.000094 |
| % Idiosyncratic volatility | 7398.73 | 1 | 22.75 | 0.000002 |
| % T-bill rate | 7406.89 | 1 | 30.91 | < 0.000001 |
| % Idiosyncratic volatility (spline term) | 7413.60 | 3 | 41.62 | < 0.000001 |
| % Distance-to-default | 7418.65 | 1 | 42.67 | < 0.000001 |
| % Net income / size | 7423.10 | 1 | 47.12 | < 0.000001 |
| % Total liabilities / size | 7457.40 | 1 | 81.42 | < 0.000001 |
| % Log excess return | 7732.36 | 1 | 356.38 | < 0.000001 |
Rows are models without the given term. The spline terms are orthogonal to the linear term. P-values are based on likelihood ratio test.
Gray area is pointwise 95% confidence intervals. The line is the predicted distress rate. Dots are realised distress rates. Black dots are not covered by confidence intervals.
Model is estimated up to the year prior to the forecasting year.
Starting point is a time-varying intercept as in Duffie et al. (2009).
\[ \begin{align*} g(E(y_{it}|\vec x_{it}, \vec z_t, u_t)) &= \vec\beta^\top\vec x_{it}+ \vec \gamma^\top \vec z_t + u_t \\ u_t &\sim N(\theta u_{t - 1}, \sigma^2) \end{align*} \]
where \(g\) is the inverse complementary-log-log function.
It may be that do use \(\Delta t\) close to zero in their discrete approximation but it appears that they use \(\Delta t = 1\) month when you look at Appendix B and C. In that case, one could argue that a log-likelihood of Poisson process with a log-offset (the continuous version) should not be used but rather the inverse complementary-log-log function (the discrete version). It presumably does not matter.
See the R package at CRAN.R-project.org/package=dynamichazard for implementation details.
| Df | AIC | |
|---|---|---|
| Model without frailty | 17 | 7378 |
| Random intercept | 19 | 7345 |
| Random intercept and log relative size | 22 | 7229 |
| Random intercept, log relative size, and industry dummies | 25 | 7238 |
Estimates are \((\hat\theta, \hat\sigma) = (0.8912, 0.1484)\).
Investigate reduced capital requirements of SMBs in Basel II/III as in Jensen, Lando, and Medhat (2017).
Under the IRB approach for corporate credits, banks will be permitted to separately distinguish exposures to SME borrowers (defined as corporate exposures where the reported sales for the consolidated group of which the firm is a part is less than €50 million) from those to large firms. A firm-size adjustment … is made to the corporate risk weight formula for exposures to SME borrowers.
Article 273 in Basel Committee on Banking Supervision (2006).
\[ \begin{align*} g(E(y_{it}|\vec x_{it}, \vec z_t, \vec u_t, o_{it})) &= \vec\beta^\top\vec x_{it}+ \vec \gamma^\top \vec z_t + u_{I,t} +u_{R,t} o_{it} \\ \begin{pmatrix} u_{I,t} \\ u_{R,t} \end{pmatrix} &\sim N\left( \begin{pmatrix} \theta_Iu_{I,t-1} \\ \theta_Ru_{R,t-1} \end{pmatrix}, Q\right) \end{align*} \]
| Df | AIC | |
|---|---|---|
| Model without frailty | 17 | 7378 |
| Random intercept | 19 | 7345 |
| Random intercept and log relative size | 22 | 7229 |
| Random intercept, log relative size, and industry dummies | 25 | 7238 |
Estimates are
\[ \begin{align*} \begin{pmatrix} \hat\theta_I \\ \hat\theta_R \end{pmatrix} &= \begin{pmatrix} 0.9480 \\ 0.9793 \end{pmatrix} & \hat Q&= \hat V\hat C\hat V\\ \hat V&=\begin{pmatrix} 0.2446 & \cdot \\ \cdot & 0.0500 \end{pmatrix} & \hat C&=\begin{pmatrix} 1 & 0.9003 \\ 0.9003 & 1 \end{pmatrix} \end{align*} \]
Points are means for distressed firms and crosses are the means of the
whole sample. The two lines are smoothing splines.
| Model without frailty | Model with frailty | |
|---|---|---|
| Intercept | -9.8216 | -10.1538 |
| Retained Earnings / size | 0.0923 | 0.0850 |
| Net income / size | ||
| Total liabilities / size | 1.1879 | 1.1457 |
| Current ratio | -0.0979 | -0.1366 |
| Idiosyncratic volatility | ||
| Log excess return | -1.3661 | -1.3241 |
| Relative log market size | -0.0467 | 0.0397 |
| Distance-to-default | -0.3909 | -0.3711 |
| Log market return | 0.4106 | 0.5064 |
| T-bill rate | -8.3157 | -4.2765 |
| Spline Net income / size | ||
| Spline Idiosyncratic volatility |
Lando et al. (2013) find a time-varying log pledgeable assets effects in a non-parametric Aalen model.
Azizpour, Giesecke, and Schwenkler (2018) uses exponentially weighted log past defaulted debt in aggregate distress prediction.
Advocated by Chava and Jarrow (2004) and Chava, Stefanescu, and Turnbull (2011).
No evidence of industry effects in the model without frailty.
Use dummies from Chava and Jarrow (2004):
\[ \begin{align*} g(E(y_{it}|\vec x_{it}, \vec z_t, \vec u_t, \vec o_{it})) &= \vec\beta^\top\vec x_{it}+ \vec \gamma^\top \vec z_t + \vec u_{t}^\top\vec o_{it} \\ \begin{pmatrix} u_{I_1,t} \\u_{I_2,t} \\u_{I_3,t} \\ u_{R,t} \end{pmatrix} &\sim N\left( \begin{pmatrix} \theta_Iu_{I_1,t-1} \\ \theta_Iu_{I_2,t-1} \\ \theta_Iu_{I_3,t-1} \\ \theta_Ru_{R,t-1} \end{pmatrix}, Q\right) \end{align*}, \qquad Q=VCV \]
\[ V=\begin{pmatrix} \sigma_{I_1} & \cdot & \cdot & \cdot \\ \cdot & \sigma_{I_2} & \cdot & \cdot \\ \cdot & \cdot & \sigma_{I_3} & \cdot \\ \cdot & \cdot & \cdot & \sigma_R \end{pmatrix}, \qquad C = \begin{pmatrix} 1 & \rho_I & \rho_I & \rho_{IR} \\ \rho_I & 1 & \rho_I & \rho_{IR} \\ \rho_I & \rho_I & 1 & \rho_{IR} \\ \rho_{IR} & \rho_{IR} & \rho_{IR} & 1 \end{pmatrix} \]
| Df | AIC | |
|---|---|---|
| Model without frailty | 17 | 7378 |
| Random intercept | 19 | 7345 |
| Random intercept and log relative size | 22 | 7229 |
| Random intercept, log relative size, and industry dummies | 25 | 7238 |
\[ \begin{align*} (\hat\theta_I, \hat\theta_R) &= (0.9062, 0.9616) \\ (\hat\sigma_{I_1}, \hat\sigma_{I_2}, \hat\sigma_{I_3},\hat\sigma_R) &= (0.2998, 0.3484, 0.3049, 0.0607) \\ (\hat\rho_I, \hat\rho_{IR}) &= (0.9029, 0.9065) \end{align*} \]
The log-likelihood in the last model is actually smaller as is evident by the AIC. Though, it does seem like the EM is close to having converged in the latter model.
Compare with Kwon and Lee (2018).
Compare with Chen and Wu (2014).
Rarely focused on. One example is Berg (2007).
Partial association between size of the firm and distress is non-constant through time.
Weak evidence for partial industry effects.
Document importance. Could make model for \((\vec x_{it}, \vec z_t, \vec o_{it})\) as in Duffie, Saita, and Wang (2007), Duffie et al. (2009), Chen and Wu (2014), and Qi, Zhang, and Zhao (2014). This may lead to poor performance as documented in Duan, Sun, and Wang (2012).
Alternatively, forgo the the covariate prediction problem and make a 1 year distress model.
Explore other time-varying coefficients.
Have no standard errors for effects. Compute observed information matrix with numerical differentiation or the methods in e.g., Cappé, Moulines, and Ryden (2005) or Poyiadjis, Doucet, and Singh (2011).
Potential issue with Moody’s data.
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